Title:Enumerating Colorings, Tensions and Flows in Cell Complexes

Abstract: We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for
some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
$X$.

Comments:

28 pages, 3 figures. Final version, to appear in J. Combin. Theory Series A